holds for any real number k . In this paper, we focus on the case of q write

10 , and we
10,k

Keywords
F.Smarandache, sum of base 10, Stirling number

(x ) , and use combinational methods

b(m)

B10 (m)

and

k

(x )

(x )

for

to deduce some exact formulas for it, which related the Stirling numbers.



convenience. In problem 21 of book [3], Professor F.Smaradache asks us to study the properties of sequence {b(m)} . The problem is interesting because it can help us to find some new distribution properties of the base 10 digits. R. E. Kennedy[5] and C. Cooper[6] obtained repectively for any fixed k N , then

1. Introduction
Let q be a positive integer. It is well known that any non-negetive integer m can be uniquely written in the base q-digits as:

(x ) k
and

x

9 logk x 2

k

O x log

k

1 3

x ,

(Equ. 6)

m

10n n

n

10n 1

1

10 1
i n

, 0

(Equ. 1)

10 , 0 n i consider the so-called digital sum

where
Bq (m)

0,0

1 .It is quite to

k

(10 )

n

9 x n k 10n 2

k

O n k 110n .

(Equ. 7)

n

n 1

. 0

(Equ. 2)

We are interested in the average behavior of the form
q ,k

(x )

Bqk (m),

(Equ. 3)

Professor zhang deduced two exact calculating formulas ( x ) in [6]. k The main purpose of this paper is to study the calculating problem of (x ), and use combinational k methods to deduce some exact formulas for it, which related the first Stirling numbers s(n, k ) and the second Stirling numbers S (n, k ) (See [7]).

where k is a fixed real number. In most cases, we are more interested in the cases of q 2 and q 10 . For the former one, known as the binary digital sum, it is proved by K. B. Stolarsky [1] that

2. Main Results
Let

(x ) 2,k

log x x log 2

k

1

O

log log x log x

w(n, r ) : #{b(m), m
(Equ. 4)

10n

1},

(Equ. 8)

that is, w(n, r ) denotes all the numbers of non-negative integer m which satisfies b(m) r in the base 10 digits,it is easy to know that 0 r 9n . Then the generating function of w(n, r ) can be writen as:
9n

holds for any k Coquet [2] that

0 . Subsequently, it is improved by J.

This work was supported by the Weinan Teachers University. Zhibin Ren is with P.R. China Weinan Teachers University of Mathematics and Information Science. (email: zb r@wntc.edu.cn)
.